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Useful hypotheses: - Rely on specifying - null hypothesis (Ho) - alternate hypothesis (Ha)
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Together Ho and Ha encompass all possible outcomes: - For Example:
- Ho: µ=0, Ha: µ ≠ 0 - mean equals 0 or mean does not equal 0 - Ho: µ=3700, Ha: µ ≠ 3700 - mean equals 3700 or mean does not equal 3700 - Ho: µ1 = µ2, Ha: µ1 ≠ µ2 - mean of population 1 equals mean of population 2 or it does not - Ho: µ > 0, Ha: µ ≤ 0 - can be directional mean is greater than 0 or mean is not equal or less than 0
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Tests assess likelihood of the null hypothesis being true
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Hypothesis tests
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Statistical test results: - p = 0.3 means that if I repeated the study 100 times, I would get this (or more extreme) result due to chance 30 times - p = 0.03 means that if I repeated the study 100 times, I would get this (or more extreme) result due to chance 3 times
Which p-value suggests Ho likely false?
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Statistical test results:
At what point reject Ho? - p < 0.05 conventional “significance threshold” (α = alpha or p value) - p < 0.05 means: - if Ho is true and we repeated the study 100 times - we would get this (or more extreme) result less than 5 times due to chance
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Statistical test results: - α is the rate at which we will reject a true null hypothesis (Type I error rate) - Lowering α will lower likelihood of incorrectly rejecting a true null hypothesis (e.g., 0.01, 0.001)
*Both Hs and α are specified **BEFORE collection of data and analysis*
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Traditionally α=0.05 is used as a cut off for rejecting null hypothesis
There is nothing magical about 0.05 - actual p-values need to be reported - also need to decide prior to study
| p-value range | Interpretation |
|---|---|
| P > 0.10 | No evidence against Ho - data appear consistent with Ho |
| 0.05 < P < 0.10 | Weak evidence against the Ho in favor of Ha |
| 0.01 < P < 0.05 | Moderate evidence against Ho in favor of Ha |
| 0.001 < P < 0.01 | Strong evidence against Ho in favor of Ha |
| P < 0.001 | Very strong evidence against Ho in favor of Ha |
Fisher:
p-value as informal measure of discrepancy betwen data and Ho
“If p is between 0.1 and 0.9 there is certainly no reason to suspect the hypothesis tested. If it is below 0.02 it is strongly indicated that the hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at .05 …”
General procedure for H testing:
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General procedure for H testing: - Collect data - Perform test - If p-value < α, conclude Ho is likely false and reject it - If p-value > α, conclude no evidence Ho is false and retain it
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Recall… - Major goal of statistics: inferences about populations from samples… and assign degree of confidence to inferences - Statistical H-testing: formalized approach to inference - Relies on specifying null hypothesis (Ho) and alternate hypothesis (Ha) - Tests assess likelihood of the null hypothesis being true - Expressed as p-value: probability of obtaining sample value of statistic (or more extreme one) if Ho is true
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Recall hospital example - Probability of getting sample like A (with ȳ at least as far away from 3700 as 3500)? - p(ȳ ≤ 3500 or ȳ ≥ 3900)
What about - 1-tailed or 2-tailed test?
Can solve using SND and z-scores
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z= (3500-3700)/410 = -0.48
But- usually can’t use z!
Can use t-distribution instead…
Let’s rephrase the question somewhat: What is probability that sample A (ȳ=3500, s=676) is from population with µ = 3700 (and σ = ?)?
This is a 1-sample test - 1-sample tests ask whether - µ is different than some number (e.g., 0 or 3700 or whatever) - based on sample - Another way: - probability this sample came from population with certain µ (mean) (e.g., 3700)
Common approach is one-sample t-test, which uses t statistic:
- St is value of statistic of interest from sample (e.g., ȳ) - θ is population value if Ho is true (e.g., µ=0) - SSt is standard error of the mean (s/√n)
What does equation tell us about relationships between variables?
test statistic (ts) is calculated by taking difference between observed statistic and value you’d expect under Ho then dividing by the standard error.
This standardization allows you to measure how many “standard errors away” your result is from what would be expected if the null hypothesis were true.
Process:
Specify Ho (e.g., µ = 0) and HA (e.g., µ ≠ 0)
Specify significance level (e.g., α = 0.05)
Take sample from population
Calculate:
If sample from pop with mean or µ = 0 then t will be close to 0
|Large| t values are more likely if Ho false
Compare t with t-distribution: - if t is further than t at specified significance level (p = 0.05) (σ) - t has less than 5% chance of coming from null distribution
Is birth weight in population significantly different than 3700 g? - Specify Ho: µ = 3700 and Ha: µ ≠ 3700 - Specify significance level (α = 0.05) - Take sample from population: ȳ = 3500, s = 676
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Is this one-tailed or two-tailed question?
Calculate t = (3500 – 3700) / (676/ √ 20)
t = -1.32
Compare with t-distribution for df=19
~0.10 of t distribution left of t=-1.32 and ~0.1 right of 1.32 (~0.2 overall)
Probability of getting a sample as extreme (or more) as this is ~0.2
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